{
  "id": "2407.11783",
  "version": "v1",
  "published": "2024-07-16T14:39:27.000Z",
  "updated": "2024-07-16T14:39:27.000Z",
  "title": "Uniform exclude distributions of Sidon sets",
  "authors": [
    "Darrion Thornburgh"
  ],
  "comment": "21, 5 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A Sidon set $S$ in $\\mathbb{F}_2^n$ is a set such that the pairwise sums of distinct points are all distinct. The exclude points of a Sidon set $S$ are the sums of three distinct points in $S$, and the exclude multiplicity of a point in $\\mathbb{F}_2^n \\setminus S$ is the number of such triples in $S$ it is equal to. We call the function $d_S \\colon \\mathbb{F}_2^n \\setminus S \\to \\mathbb{Z}_{\\geq 0}$ taking points in $\\mathbb{F}_2^n \\setminus S$ to their exclude multiplicity the exclude distribution of $S$. We say that $d_S$ is uniform on $\\mathcal{P}$ if $\\mathcal{P}$ is an equally-sized partition $\\mathcal{P}$ of $\\mathbb{F}_2^n \\setminus S$ such that $d_S$ takes the same values an equal number of times on every element of $\\mathcal{P}$. In this paper, we use APN plateaued functions with all component functions unbalanced to construct Sidon sets $S$ in $(\\mathbb{F}_2^n)^2$ whose exclude distributions are uniform on natural partitions of $(\\mathbb{F}_2^n)^2 \\setminus S$ into $2^n$ elements. We use this result and a result of Carlet to determine exactly what values the exclude distributions of the graphs of the Gold and Kasami functions take and how often they take these values.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-16T14:39:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "uniform exclude distributions",
      "exclude multiplicity",
      "distinct points",
      "construct sidon sets",
      "natural partitions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}